optim function

Chapter 1

Optimization usingoptim() in R

An in-class activity to apply Nelder-Mead and Simulated Annealing in

optim() for a variety of bivariate functions.

# SC1 4/18/2013

# Everyone optim()!

# The goal of this exercise is to minimize a function using R's optim().

# Steps:

# 0. Break into teams of size 1 or 2 students.

# 1. Each team will choose a unique function from this list:

# Test functions for optimization

# http://en.wikipedia.org/wiki/Test_functions_for_optimization

# 1a. Claim the function by typing your names into the function section below.

# 1b. Click on "edit" on Wikipedia page to copy latex math for function

# and paste between dollar signs $f(x)$

# 2. Following my "Sphere function" example:

# 2a. Define function()

# 2b. Plot the function

# 2c. Optimize (minimize) the function

# 2d. Comment on convergence

# 3. Paste your work into your function section.

# 4. I'll post this file on the website for us all to enjoy, as well as create

# a lovely pdf with images of the functions.

2 Optimization using optim() in R

1.1 Sphere function

f (x) =∑n

i=1 x2i

########################################

# Sphere function

# Erik Erhardt

# $f(\boldsymbol{x}) = \sum_{i=1}^{n} x_{i}^{2}$

# name used in plot below

f.name <- "Sphere function"

# define the function

f.sphere <- function(x) {# make x a matrix so this function works for plotting and for optimizing

x <- matrix(x, ncol=2)

# calculate the function value for each row of x

f.x <- apply(x^2, 1, sum)

# return function value

return(f.x)

}

# plot the function

# define ranges of x to plot over and put into matrix

x1 <- seq(-10, 10, length = 101)

x2 <- seq(-10, 10, length = 101)

X <- as.matrix(expand.grid(x1, x2))

colnames(X) <- c("x1", "x2")

# evaluate function

y <- f.sphere(X)

# put X and y values in a data.frame for plotting

df <- data.frame(X, y)

# plot the function

library(lattice) # use the lattice package

wireframe(y ~ x1 * x2 # y, x1, and x2 axes to plot

, data = df # data.frame with values to plot

, main = f.name # name the plot

, shade = TRUE # make it pretty

, scales = list(arrows = FALSE) # include axis ticks

, screen = list(z = -50, x = -70) # view position

)

# optimize (minimize) the function using Nelder-Mead

1.1 Sphere function 3

out.sphere <- optim(c(1,1), f.sphere, method = "Nelder-Mead")

out.sphere

## $par

## [1] 3.754e-05 5.179e-05

##

## $value

## [1] 4.092e-09

##

## $counts

## function gradient

## 63 NA

##

## $convergence

## [1] 0

##

## $message

## NULL

# optimize (minimize) the function using Simulated Annealing

out.sphere <- optim(c(1,1), f.sphere, method = "SANN")

out.sphere

## $par

## [1] 0.0001933 -0.0046280

##

## $value

## [1] 2.146e-05

##

## $counts


## 10000 NA

##

## $convergence

## [1] 0

##

## $message

## NULL

###

# comments based on plot and out.*


# The unique minimum was found within tolerance.

## values of x1 and x2 at the minimum

# $par

# [1] 3.754010e-05 5.179101e-05

#

## value of the function at the minimum

# $value

# [1] 4.091568e-09

#

## convergence in 63 iterations

# $counts

# function gradient

# 63 NA

#

## 0 = convergence successful

# $convergence

# [1] 0

#

## no news is good news

# $message

# NULL

Sphere function

−10

−5

0

5

10 −10

−5

05

10

0

50

100

150

200

x1x2

y

1.2 Sphere function with stochastic noise 5

1.2 Sphere function with stochastic noise

########################################

# Sphere function with stochastic noise

# Christian Gunning


f.name <- "Sphere function with stochastic noise at each iteration"


f.sphere1 <- function(x) {# make x a matrix so this function works for plotting and for optimizing



# f.x <- apply(x, 1, function(y) {ret<- sum(y^2) })f.x <- apply(x, 1, function(y) {ret<- sum(y^2)+rnorm(1,mean=1,sd=abs(mean(y))^(1/10))})# return function value

return(f.x)

}

# plot the function


x1 <- seq(-10, 10, length = 101)

x2 <- seq(-10, 10, length = 101)



# evaluate function

y <- f.sphere1(X)



# plot the function








)


out.NM <- optim(c(1,1), f.sphere1, method = "Nelder-Mead")

out.NM


## $par

## [1] 0.875 1.150

##

## $value

## [1] 0.2255

##

## $counts


## 321 NA

##

## $convergence

## [1] 10

##

## $message

## NULL


out.sann <- optim(c(1,1), f.sphere1, method = "SANN")

out.sann

## $par

## [1] -0.7529 -0.3134

##

## $value

## [1] -1.036

##

## $counts


## 10000 NA

##

## $convergence

## [1] 0

##

## $message

## NULL

1.2 Sphere function with stochastic noise 7

Sphere function with stochastic noise at each iteration

−10

−5

0

5

10 −10

−5

05

10

0

50

100

150

200

x1x2

y


1.3 Rosenbrock function

########################################

# Rosenbrock function

# Mary Rose Paiz


f.name <- "Rosenbrock Function"


f.rosenbrock <- function(x) {

x1 <- x[,1]

x2 <- x[,2]

# calculating f.x

term1 <- (x2 - (x1)^2)^2

term2 <- (x1 - 1)^2

f.x <- (100*term1 + term2)


return(f.x)

}

# plot the function


x1 <- seq(-1.5, 2.0, length = 101)

x2 <- seq(.5, 3.0, length = 101)



# evaluate function

y <- f.rosenbrock(X)



# plot the function








)

1.3 Rosenbrock function 9


out.rosenbrock <- optim(c(1,1), f.rosenbrock, method = "Nelder-Mead")

## Error: incorrect number of dimensions

out.rosenbrock

## Error: object ’out.rosenbrock’ not found


out.rosenbrock <- optim(c(1,1), f.rosenbrock, method = "SANN")

## Error: incorrect number of dimensions

out.rosenbrock

## Error: object ’out.rosenbrock’ not found

Rosenbrock Function

−1.5

−1.0

−0.5

0.0

0.5

1.0

1.5

2.00.5

1.01.5

2.02.5

3.0

200

400

600

800

1000

1200

x1

x2

y


1.4 Beale’s function

f (x, y) = (1.5− x + xy)2 +(2.25− x + xy2

)2+(2.625− x + xy3

)2.

########################################

# Beale's function

# Alvaro

#$$f(x,y) = \left( 1.5 - x + xy \right)^{2} + \left( 2.25 - x + xy^{2}\right)^{2} + \left(2.625 - x+ xy^{3}\right)^{2}.$$

#Minimum:

#$$f(3, 0.5) = 0\\#-4.5 \le x,y \le 4.5$$


f.name <- "Beale's function"


f.beale <- function(mx) {mx <- matrix(mx, ncol=2)

x<- mx[,1]

y<- mx[,2]

f.x<- (1.5 - x +x*y)^2 + (2.25-x+(x*y)^2)^2 + (2.625-x+(x*y)^3)^2

return(f.x)

}

# plot the function


x1 <- seq(-4.5, 4.5, length = 101)

x2 <- seq(-4.5, 4.5, length = 101)



# evaluate function

y <- log10(f.beale(X))



# plot the function





1.4 Beale’s function 11



, screen = list(z = 0, x = 0) # view position

)


out.beale <- optim(c(1,1), f.beale, method = "Nelder-Mead")

out.beale

## $par

## [1] 2.4814 0.2284

##

## $value

## [1] 0.286

##

## $counts


## 83 NA

##

## $convergence

## [1] 0

##

## $message

## NULL


out.beale <- optim(c(1,1), f.beale, method = "SANN")

out.beale

## $par

## [1] 2.4830 0.2269

##

## $value

## [1] 0.2861

##

## $counts


## 10000 NA

##

## $convergence

## [1] 0

##

## $message

## NULL


###

# comments based on plot and out.*

# The unique minimum was found within tolerance.

Beale's function

−4 −2 0 2 4

−4

−2

0

2

4

02

46

x1

x2

y

1.5 Goldstein-Price function 13

1.5 Goldstein-Price function

f (x, y) =(

1 + (x + y + 1)2(19− 14x + 3x2 − 14y + 6xy + 3y2

))(30 + (2x− 3y)2

(18− 32x + 12x2 + 48y − 36xy + 27y2

))########################################

# Goldstein-Price function

# Barnaly Rashid

#GoldsteinPrice function:

#$f(x,y) = \left(1+\left(x+y+1\right)^{2}\left(19-14x+3x^{2}-14y+6xy+3y^{2}\right)\right)\left(30+\left(2x-3y\right)^{2}\left(18-32x+12x^{2}+48y-36xy+27y^{2}\right)\right)$

f.name <- "Goldstein-Price function"


f.goldprice <- function(x1x2) {# calculate the function value for x1 and x2

x1x2 <- matrix(x1x2,ncol=2)

a <- 1+(x1x2[,1]+x1x2[,2]+1)^2*(19-14*x1x2[,1]+3*x1x2[,1]^2-14*x1x2[,2]+6*x1x2[,1]*x1x2[,2]+3*x1x2[,2]^2)

b <- 30 + (2*x1x2[,1]-3*x1x2[,2])^2*(18-32*x1x2[,1]+12*x1x2[,1]^2+48*x1x2[,2]-36*x1x2[,1]*x1x2[,2]+27*x1x2[,2]^2)

f.x <- a*b


return(f.x)

}

# matrix(x1x2,ncol=2)

#plot the function

# define ranges of x to plot over

x1 <- seq(-1.5, 1.5, length = 101)

x2 <- seq(-1.5, 1.5, length = 101)



#y <- f.goldprice(X[,1],X[,2])

y <- f.goldprice(X)











)


out.gold <- optim(c(0,-1), f.goldprice, method = "Nelder-Mead")

out.gold

## $par

## [1] 0 -1

##

## $value

## [1] 3

##

## $counts


## 57 NA

##

## $convergence

## [1] 0

##

## $message

## NULL


out.gold <- optim(c(0,-1), f.goldprice, method = "SANN")

out.gold

## $par

## [1] 0 -1

##

## $value

## [1] 3

##

## $counts


## 10000 NA

##

1.5 Goldstein-Price function 15

## $convergence

## [1] 0

##

## $message

## NULL

Goldstein−Price function

−1.5−1.0

−0.50.0

0.51.0

1.5 −1.5−1.0

−0.50.0

0.51.0

1.5

1e+05

2e+05

3e+05

x1x2

y


1.6 Booth’s function

f (x) = (x + 2y − 7)2 + (2x + y − 5)2

########################################

# Booth's function

# Olga Vitkovskaya

# $f(\boldsymbol{x}) = \(x + 2y -7)^{2}+(2x + y -5)^{2}$


f.name <- "Booth's function"


f.booths <- function(xy) {

# make x a matrix so this function works for plotting and for optimizing

xy <- matrix(xy, ncol=2)


f.row <- function(this.row) {(this.row[1] + 2 * this.row[2] -7)^2 + (2 * this.row[1] + this.row[2] -5)^2

}f.x <- apply(xy, 1, f.row)


return(f.x)

}

# plot the function


x.plot <- seq(-10, 10, length = 101)

y.plot <- seq(-10, 10, length = 101)

grid.plot <- as.matrix(expand.grid(x.plot, y.plot))

colnames(grid.plot) <- c("x", "y")

# evaluate function

z.plot <- f.booths(grid.plot)

# put X, y and z values in a data.frame for plotting

df <- data.frame(grid.plot, z.plot)

# plot the function


1.6 Booth’s function 17

p <- wireframe(z.plot ~ x * y # y, x1, and x2 axes to plot






)

plot(p)


out.booth1 <- optim(c(1,1), f.booths, method = "Nelder-Mead")

out.booth1

## $par

## [1] 0.9999 3.0001

##

## $value

## [1] 4.239e-08

##

## $counts


## 69 NA

##

## $convergence

## [1] 0

##

## $message

## NULL


out.booth2 <- optim(c(1,1), f.booths, method = "SANN")

out.booth2

## $par

## [1] 1.004 3.003

##

## $value

## [1] 0.0002215

##

## $counts


## 10000 NA

##


## $convergence

## [1] 0

##

## $message

## NULL

Booth's function

−10

−5

0

5

10 −10

−5

05

10

0

500

1000

1500

2000

2500

xy

z.plot


1.7 Booth’s function

f (x, y) = (x + 2y − 7)2 + (2x + y − 5)2 .

########################################

# Booth's function

# {Katherine Freeland)

# Booth's Function: $f(x,y) = \left( x + 2y -7\right)^{2} + \left(2x +y - 5\right)^{2}.\quad$# Minimum: $f(1,3) = 0</math>, for <math>-10 \le x,y \le 10</math>.$

f.booth <- function(xy){xy <- matrix(xy, ncol=2)

f.x <- ((xy[,1] + (2*xy[,2]) - 7)^2) + ((2*xy[,1]+ xy[,2]-5)^2)

return(f.x)

}

x <- seq(-5, 5, length=101)

y <- seq(-5, 5, length=101)

mat <- as.matrix(expand.grid(x, y))

colnames(mat) <- c("x", "y")

f.x <- f.booth(mat)

df <- data.frame(mat, f.x)


wireframe(f.x ~ x * y # f.x, x, and y axes to plot


, main = "Booth Function" # name the plot




)


out.booth <- optim(c(1,1), f.booth, method = "Nelder-Mead")

out.booth

## $par

## [1] 0.9999 3.0001

##

## $value

## [1] 4.239e-08


##

## $counts


## 69 NA

##

## $convergence

## [1] 0

##

## $message

## NULL

out.booth2 <- optim(c(1,1), f.booth, method = "SANN")

out.booth2

## $par

## [1] 1.000 3.002

##

## $value

## [1] 1.658e-05

##

## $counts


## 10000 NA

##

## $convergence

## [1] 0

##

## $message

## NULL


Booth Function

−4−2

02

4 −4−2

02

4

0

200

400

600

800

xy

f.x


1.8 Bukin function N. 6

f (x, y) = 100√|y − 0.01x2| + 0.01 |x + 10|

########################################

# Bukin function N. 6

# {Zhanna G.}

# $f(x,y) = 100\sqrt{\left|y - 0.01x^{2}\right|} + 0.01 \left|x+10 \right|$

f.name <- "Bukin_6 function"


f.bukin <- function(xy) {xy <- matrix(xy, ncol=2)


f.xy <- 100*sqrt(abs(y-0.01*(x)^2)) + 0.01*abs(x+10)


return(f.xy)

}

x <- seq(-15, -5, length = 101)

y <- seq(-3, 3, length = 101)

X <- as.matrix(expand.grid(x, y))

#X

colnames(X) <- c("x", "y")

Z <- f.bukin(X)

#Z

df <- data.frame(X, Z)

#head(df)

# plot the function


wireframe(Z ~ x * y # y, x, and z axes to plot






)

1.8 Bukin function N. 6 23

Bukin_6 function

−14−12

−10−8

−6−3

−2−1

01

23

50

100

150

200

xy

Z


1.9 Ackley’s function

f (x, y) = −20 exp(−0.2

√0.5 (x2 + y2)

)−exp (0.5 (cos (2πx) + cos (2πy)))+

20 + e.

########################################

# Ackley's function

# Rob Hoy

# $<math>f(x,y) = -20\exp\left(-0.2\sqrt{0.5\left(x^{2}+y^{2}\right)}\right)-\exp\left(0.5\left(\cos\left(2\pi x\right)+\cos\left(2\pi y\right)\right)\right) + 20 + e.\quad</math>$


f.name <- "Ackley's function"


f.ackley <- function(X) {m <- matrix(X, ncol=2)

# calculate the function value

t1 <- (-20*(exp(-.2*sqrt(.5*(m[,1]^2+m[,2]^2)))))

t2 <- (exp(.5*(cos(2*pi*m[,1]) + cos(2 * pi * m[,2]))))

z <- t1 - t2 + 20 + exp(1)


return(z)

}

# define ranges of x and y to plot

x <- seq(-10, 10, length = 101)

y <- seq(-10, 10, length = 101)

# make x and y a matrix, plotting and opt.



# evaluate function

z <- f.ackley(X)

# Create dataframe for graphing

df.ack <-data.frame(X,z)

# plot the function


wireframe(z ~ x * y # z, x, and y axes to plot

, data = df.ack # data.frame with values to plot


1.9 Ackley’s function 25




)


out.ackley1 <- optim(c(-1,1), f.ackley, method = "Nelder-Mead")

out.ackley1

## $par

## [1] -0.9685 0.9685

##

## $value

## [1] 3.574

##

## $counts


## 45 NA

##

## $convergence

## [1] 0

##

## $message

## NULL


out.ackley2 <- optim(c(1,1), f.ackley, method = "SANN")

out.ackley2

## $par

## [1] 0.001159 0.003890

##

## $value

## [1] 0.01192

##

## $counts


## 10000 NA

##

## $convergence

## [1] 0

##

## $message

## NULL


#The first one was faster, but it appears to me that the second one is actually the more accurate.

Ackley's function

−10

−5

0

5

10 −10

−5

05

10

5

10

15

xy

z

1.10 Matyas function 27

1.10 Matyas function

f (x, y) = 0.26(x2 + y2

)− 0.48xy.

########################################

# Matyas function

# Josh Nightingale

# $f(x,y) = 0.26 \left( x^{2} + y^{2}\right) - 0.48 xy.$


f.name <- "Matyas function"


f.matyas <- function(XY) {# make x a matrix so this function works for plotting and for optimizing

XY <- matrix(XY, ncol=2)

x <- XY[,1]

y <- XY[,2]


f.xy <- (0.26 * (x^2 + y^2)) - (0.48 * x * y)

return(f.xy)

}

# plot the function


x <- seq(-10, 10, length = 101)

y <- seq(-10, 10, length = 101)

XY <- as.matrix(expand.grid(x, y))

colnames(XY) <- c("x", "y")

# evaluate function

z <- f.matyas(XY)


df <- data.frame(XY, z)

# plot the function


wireframe(z ~ x * y # z, x, and y axes to plot





#, screen = list(z = 3, x = 5) # view position

)



out.matyas <- optim(c(1,1), f.matyas, method = "Nelder-Mead")

out.matyas

## $par

## [1] 8.526e-05 7.856e-05

##

## $value

## [1] 2.796e-10

##

## $counts


## 69 NA

##

## $convergence

## [1] 0

##

## $message

## NULL


out.matyas <- optim(c(1,1), f.matyas, method = "SANN")

out.matyas

## $par

## [1] 0.02710 0.01713

##

## $value

## [1] 4.442e-05

##

## $counts


## 10000 NA

##

## $convergence

## [1] 0

##

## $message

## NULL

1.10 Matyas function 29

Matyas function

−10

−5

0

5

10

−10

−5

0

5

100

20

40

60

80

100

xy

z


1.11 Levi function N. 13

f (x, y) = sin2 (3πx)+(x− 1)2(1 + sin2 (3πy)

)+(y − 1)2

(1 + sin2 (2πy)

).

########################################

# Levi function N. 13

# Claire L

# $f(x,y) = \sin^{2}\left(3\pi x\right)+\left(x-1\right)^{2}\left(1+\sin^{2}\left(3\pi y\right)\right)+\left(y-1\right)^{2}\left(1+\sin^{2}\left(2\pi y\right)\right).\quad$


f.name <- "Levi function"


f.levi <- function(X) {# make x a matrix so this function works for plotting and for optimizing

# x <- matrix(x, ncol=1)

# y <- matrix(y, ncol=1)

X <- matrix(X, ncol=2)


f.xy <- (sin(3*pi*X[,1]))^2 + ((X[,1]-1)^2)*(1+(sin(3*pi*X[,2]))^2) + ((X[,2]-1)^2)*(1+(sin(2*pi*X[,2]))^2)


return(f.xy)

}

# plot the function


x <- seq(-5, 5, length = 101)

y <- seq(-5, 5, length = 101)



# evaluate function

z <- f.levi(X)

# put X and y and z values in a data.frame for plotting

df <- data.frame(X,z)

# plot the function

#It works! :)


wireframe(z ~ x * y





1.11 Levi function N. 13 31


)


out.levi <- optim(c(1,1), f.levi, method = "Nelder-Mead", )

out.levi

## $par

## [1] 1 1

##

## $value

## [1] 1.35e-31

##

## $counts


## 103 NA

##

## $convergence

## [1] 0

##

## $message

## NULL


out.levi <- optim(c(1,1), f.levi, method = "SANN")

out.levi

## $par

## [1] 1 1

##

## $value

## [1] 1.35e-31

##

## $counts


## 10000 NA

##

## $convergence

## [1] 0

##

## $message

## NULL


#optimize with lower and upper bounds.

out.levi <- optim(c(1,1), f.levi, method = "L-BFGS-B", lower=-1, upper=1)

out.levi

## $par

## [1] 1 1

##

## $value

## [1] 1.35e-31

##

## $counts


## 1 1

##

## $convergence

## [1] 0

##

## $message

## [1] "CONVERGENCE: NORM OF PROJECTED GRADIENT <= PGTOL"

Levi function

−4−2

02

4 −4−2

02

4

20

40

60

80

100

120

xy

z

1.12 Three-hump camel function 33

1.12 Three-hump camel function

f (x, y) = 2x2 − 1.05x4 + x6

6 + xy + y2

########################################

# Three-hump camel function

# Mohammad

# Optimization

#$f(x,y) = 2x^{2} - 1.05x^{4} + \frac{x^{6}}{6} + xy + y^{2}$#$-5\le x,y \le 5$


f.name <- "Three-hump camel function"


f.camel <- function(input) {# make x a matrix so this function works for plotting and for optimizing

input <- matrix(input, ncol=2)


f.x <- (2*input[,1]^2) - (1.05*input[,1]^4) + (input[,1]^6)/6 +

input[,1]*input[,2] + input[,2]^2;

# f.x <- apply(x^2, 1, sum)


return(f.x)

}

# plot the function


x <- seq(-5, 5, length = 101)

y <- seq(-5, 5, length = 101)



# evaluate function

z <- f.camel(X)


df <- data.frame(X, z)

# plot the function


wireframe(z ~ x * y # y, x1, and x2 axes to plot







)


out.camel <- optim(runif(2,-5,5), f.camel, method = "L-BFGS-B", lower=c(-5,-5),

upper=c(5,5))

out.camel

## $par

## [1] 6.440e-08 -1.416e-08

##

## $value

## [1] 7.583e-15

##

## $counts


## 12 12

##

## $convergence

## [1] 0

##

## $message

## [1] "CONVERGENCE: REL_REDUCTION_OF_F <= FACTR*EPSMCH"

Three−hump camel function

−4−2

02

4−4

−2

0

24

0

500

1000

1500

2000

x

y

z

1.13 Easom function 35

1.13 Easom function

f (x, y) = − cos(x) cos(y) exp(−((x− π)2 + (y − π)2))

########################################

# Easom function

# Maozhen Gong

#f(x,y)=-\cos(x)\cos(y)\exp(-((x-\pi)^2+(y-\pi)^2))

f.name<-"Easom function"

#define the function

f.easom<-function(x){# make x a matrix so this function works for plotting and for optimizing



f.x<-apply(x,1,function(x) {-prod(cos(x)/exp((x-pi)^2))})# return function value

return(f.x)

}

# plot the function


x1 <- seq(-10, 10, length = 101)

x2 <- seq(-10, 10, length = 101)



# evaluate function

y <- f.easom(X)



# plot the function









)


out.sphere <- optim(c(3,3), f.easom, method = "Nelder-Mead")

out.sphere

## $par

## [1] 3.142 3.142

##

## $value

## [1] -1

##

## $counts


## 51 NA

##

## $convergence

## [1] 0

##

## $message

## NULL


out.sphere <- optim(c(3,3), f.easom, method = "SANN")

out.sphere

## $par

## [1] 3 3

##

## $value

## [1] -0.9416

##

## $counts


## 10000 NA

##

## $convergence

## [1] 0

##

## $message

## NULL

1.13 Easom function 37

Easom function

−10

−5

0

5

10 −10

−5

05

10

−0.8

−0.6

−0.4

−0.2

0.0

x1x2

y


1.14 Cross-in-tray function

########################################

# Cross-in-tray function

1.15 Eggholder function 39

1.15 Eggholder function

f (x, y) = − (y + 47) sin(√∣∣y + x

2 + 47∣∣)− x sin

(√|x− (y + 47)|

)########################################

# Eggholder function

# Rogers F Silva

# $f(x,y) = - \left(y+47\right) \sin \left(\sqrt{\left|y + \frac{x}{2}+47\right|}\right) - x \sin \left(\sqrt{\left|x - \left(y + 47 \right)\right|}\right)$# Minimum: $f(512, 404.2319) = -959.6407$, for $-512\le x,y \le 512$.

# $f(\boldsymbol{x}) = \sum_{i=1}^{n} x_{i}^{2}$


f.name <- "Eggholder function"


f.egg <- function(x) {# make x a matrix so this function works for plotting and for optimizing



x1 = x[,1];

x2 = x[,2];

f.x <- -(x2+47)*sin(sqrt(abs(x2+x1/2+47))) - x1*sin(sqrt(abs(x1-(x2+47))))


return(f.x)

}

# plot the function


x1 <- seq(-512, 512, length = 129)

x2 <- seq(-512, 512, length = 129)



# evaluate function

y <- f.egg(X)



# plot the function









)


out.egg <- optim(c(500,400), f.egg, method = "Nelder-Mead", control = list(trace = TRUE))

## Nelder-Mead direct search function minimizer

## function value for initial parameters = -846.569207

## Scaled convergence tolerance is 1.26149e-05

## Stepsize computed as 50.000000

## BUILD 3 -76.457443 -895.756940

## LO-REDUCTION 5 -733.894449 -895.756940

## SHRINK 9 16.755533 -895.756940

## LO-REDUCTION 11 -46.997041 -895.756940

## SHRINK 15 6.846694 -895.756940

## LO-REDUCTION 17 -89.531642 -895.756940

## LO-REDUCTION 19 -601.209387 -895.756940

## LO-REDUCTION 21 -743.937706 -895.756940

## HI-REDUCTION 23 -871.318184 -895.756940

## REFLECTION 25 -892.034514 -911.383876

## SHRINK 29 -540.115854 -911.383876

## LO-REDUCTION 31 -876.357680 -911.383876

## HI-REDUCTION 33 -900.076804 -911.383876

## HI-REDUCTION 35 -905.934548 -911.383876

## EXTENSION 37 -906.836013 -918.289594

## LO-REDUCTION 39 -911.383876 -918.289594

## EXTENSION 41 -915.300166 -927.479612

## EXTENSION 43 -918.289594 -934.086287

## EXTENSION 45 -927.479612 -950.554116

## LO-REDUCTION 47 -934.086287 -950.554116

## REFLECTION 49 -949.824192 -956.159307

## LO-REDUCTION 51 -950.554116 -956.186073

## LO-REDUCTION 53 -955.918016 -956.186073

## HI-REDUCTION 55 -956.159307 -956.713849

## HI-REDUCTION 57 -956.186073 -956.775840

## HI-REDUCTION 59 -956.713849 -956.846279

## HI-REDUCTION 61 -956.775840 -956.854776

## LO-REDUCTION 63 -956.846279 -956.897279

## HI-REDUCTION 65 -956.854776 -956.900910

## HI-REDUCTION 67 -956.897279 -956.909283

## HI-REDUCTION 69 -956.900910 -956.909283

## REFLECTION 71 -956.908722 -956.911104

## HI-REDUCTION 73 -956.909283 -956.915023

1.15 Eggholder function 41

## EXTENSION 75 -956.911104 -956.917960

## HI-REDUCTION 77 -956.915023 -956.917960

## LO-REDUCTION 79 -956.916157 -956.917960

## HI-REDUCTION 81 -956.917804 -956.917960

## HI-REDUCTION 83 -956.917950 -956.918158

## HI-REDUCTION 85 -956.917960 -956.918187

## HI-REDUCTION 87 -956.918158 -956.918205

## HI-REDUCTION 89 -956.918187 -956.918215

## LO-REDUCTION 91 -956.918205 -956.918221

## Exiting from Nelder Mead minimizer

## 93 function evaluations used

out.egg

## $par

## [1] 482.4 432.9

##

## $value

## [1] -956.9

##

## $counts


## 93 NA

##

## $convergence

## [1] 0

##

## $message

## NULL


out.egg <- optim(c(500,400), f.egg, method = "SANN", control = list(trace = TRUE))

## sann objective function values

## initial value -846.569207

## iter 1000 value -965.388229

## iter 2000 value -976.124930

## iter 3000 value -976.861171

## iter 4000 value -976.910951

## iter 5000 value -976.910951

## iter 6000 value -976.910951

## iter 7000 value -976.910951

## iter 8000 value -976.910951


## iter 9000 value -976.910951

## iter 9999 value -976.910951

## final value -976.910951

## sann stopped after 9999 iterations

out.egg

## $par

## [1] 522.1 413.3

##

## $value

## [1] -976.9

##

## $counts


## 10000 NA

##

## $convergence

## [1] 0

##

## $message

## NULL

Eggholder function

−400

−200

0

200

400−400

−2000

200400

−500

0

500

1000

x1

x2

y

1.16 Holder table function 43

1.16 Holder table function

########################################

# Holder table function


1.17 McCormick function

########################################

# McCormick function

1.18 Schaffer function N. 2 45

1.18 Schaffer function N. 2

f (x, y) = 0.5 +sin2(x2−y2)−0.5

(1+0.001(x2+y2))2 .

########################################

# Schaffer function N. 2

# Yonghua

# * Schaffer function N. 2:

# :: <math>f(x,y) = 0.5 + \frac{\sin^{2}\left(x^{2} - y^{2}\right) - 0.5}{\left(1 + 0.001\left(x^{2} + y^{2}\right) \right)^{2}}.\quad</math># :Minimum: <math>f(0, 0) = 0</math>, for <math>-100\le x,y \le 100</math>.

f.name <- "Schaffer function No.2"


f.shaffer2 <- function(x) {# make x a matrix so this function works for plotting and for optimizing


f.x <- x

f.x <- cbind(x, rep(0,nrow(x)))


#for (ii in 1:nrow(x)) {

# f.x[ii,3] <- 0.5 + (sin((f.x[ii,1])^2+(f.x[ii,2])^2)-0.5)/(1+0.001*((f.x[ii,1])^2+(f.x[ii,2])^2))^2 }ret.val <- 0.5 + (sin((f.x[,1])^2+(f.x[,2])^2)-0.5)/(1+0.001*((f.x[,1])^2+(f.x[,2])^2))^2


return(ret.val)

}

# plot the function


x1 <- seq(-100, 100, length = 101)

x2 <- seq(-100, 100, length = 101)



# evaluate function

y <- f.shaffer2(X)

#colnames(y) <- c("x1", "x2", "y")











)


out.schaffer <- optim(c(100,100), f.shaffer2, method = "Nelder-Mead")

out.schaffer

## $par

## [1] 89.77 99.95

##

## $value

## [1] 0.4959

##

## $counts


## 85 NA

##

## $convergence

## [1] 0

##

## $message

## NULL


out.schaffer <- optim(c(100,100), f.shaffer2, method = "SANN")

out.schaffer

## $par

## [1] 90.6 102.6

##

## $value

## [1] 0.4961

##

## $counts

1.18 Schaffer function N. 2 47


## 10000 NA

##

## $convergence

## [1] 0

##

## $message

## NULL

Schaffer function No.2

−100

−50

0

50

100−100

−50

050

100

−0.5

0.0

0.5

x1x2

y


1.19 Schaffer function N. 4

########################################

# Schaffer function N. 4

1.20 Styblinski-Tang function 49

1.20 Styblinski-Tang function

f (x) =∑n

i=1 x4i−16x

2i+5xi

2 .

########################################

# Styblinski-Tang function

# Kathy

# $f(\boldsymbol{x}) = \frac{\sum_{i=1}^{n} x_{i}^{4} - 16x_{i}^{2} + 5x_{i}}{2}.\quad$

f.name <- "Styblinski-Tang function"

f.styblinski <- function(x) {# make x a matrix so this function works for plotting and for optimizing



f.x <- (apply((x^4 - 16 * x^2 + 5 *x) , 1, sum))/2


return(f.x)

}

# plot the function


x1 <- seq(-4.9, 5, length = 101)

x2 <- seq(-4.9, 5, length = 101)



# evaluate function

y <- f.styblinski(X)



# plot the function







, screen = list(z = 50, x = -70) # view position

)


Styblinski−Tang function

−4

−20

24

−4−2

02

4

−50

0

50

100

150

200

250

x1x2

y

optim function

Education

function section

unique function

sphere function fx

sphere function example

rs optim

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variety of bivariate

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